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Theorem sprel 42433
Description: An element of the set of all unordered pairs over a given set 𝑉 is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
Assertion
Ref Expression
sprel (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})
Distinct variable groups:   𝑉,𝑎,𝑏   𝑋,𝑎,𝑏

Proof of Theorem sprel
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 elfvex 6482 . 2 (𝑋 ∈ (Pairs‘𝑉) → 𝑉 ∈ V)
2 sprvalpw 42429 . . . 4 (𝑉 ∈ V → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
32eleq2d 2845 . . 3 (𝑉 ∈ V → (𝑋 ∈ (Pairs‘𝑉) ↔ 𝑋 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}}))
4 eqeq1 2782 . . . . . 6 (𝑝 = 𝑋 → (𝑝 = {𝑎, 𝑏} ↔ 𝑋 = {𝑎, 𝑏}))
542rexbidv 3242 . . . . 5 (𝑝 = 𝑋 → (∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏}))
65elrab 3572 . . . 4 (𝑋 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} ↔ (𝑋 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏}))
76simprbi 492 . . 3 (𝑋 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})
83, 7syl6bi 245 . 2 (𝑉 ∈ V → (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏}))
91, 8mpcom 38 1 (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  wrex 3091  {crab 3094  Vcvv 3398  𝒫 cpw 4379  {cpr 4400  cfv 6137  Pairscspr 42426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-spr 42427
This theorem is referenced by:  prssspr  42434  prsprel  42436
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